The inertial frame of reference, to which detections are stabilized, is normally one of the inertial frames used for navigation by the host Inertial Navigation System (INS). One common reference frame is the NED (North, East, Down) reference frame, defined as a local vertical, north referenced coordinate system. The D axis is directed downward, the N axis is orthogonal to the D axis and is directed- North. The E axis is orthogonal to both the N and D axes and is directed East.
The other coordinate frame of reference of interest on a host aircraft is the coordinate frame to which the INS is referenced or aligned. One commonly used reference frame is the body coordinate system (Nose, Right-wing, Down). The x or nose axis points longitudinally through the aircraft along its waterline, the y or right-wing axis is orthogonal to the x axis and is directed out the right wing. The z or down axis is orthogonal to both the x and y axes and is directed out the bottom of the aircraft. While the invention is described particularly in terms of an embodiment used in aircraft, the invention is useful in a number of other types of craft such as landcraft, watercraft, underseacraft and spacecraft.
In the right-hand coordinate system, the Euler transformation matrix [E.alpha.] for a rotation through an angle .alpha. about the z-axis is as follows: ##EQU1##
The Euler transformation matrix [E.beta.] for a rotation through an angle .beta. about the y axis is as follows: ##EQU2## And, the Euler transformation [E.UPSILON.] matrix for a rotation through an angle .UPSILON. about the x-axis is as follows: ##EQU3##
If the order of rotation were .alpha., .beta., and then .UPSILON., the total transformation [T] in three dimension would be the product of these three matrices. The order of this multiplication must be preserved. ##EQU4##
Once the angles of rotation between two coordinate systems are known, the transformation of a point in one coordinate system to the other is performed by matrix multiplication. Given two coordinates systems A and B, and knowing the Euler angles from A to B, the transformation of a point x', y', z' in B's frame back to point x, y, z in A's frame is as follows: ##EQU5##
If there were three coordinate systems, A, B and C, and the Euler transformation matrix [E.sub.AB ] from A to B and the Euler transformation matrix [E.sub.BC ] from B to C are known, then it is possible to find the Euler transformation matrix [E.sub.AC ] from A to C as follows: EQU [E.sub.AC ]=[E.sub.AB ].multidot.[E.sub.BC ] (6)
This method can be used to transform data from the sensor coordinate frame of reference to the aircraft frame of reference to the inertial coordinate system frame of reference.
This method can also be used to update the transformation matrix from the sensor coordinate frame to the inertial reference frame. In such case, the Euler matrix [E.sub.AB ] would be the previous transformation, and the Euler matrix [E.sub.BC ] would be the incremental transformation. The current transformation would just be the multiplication of the two matrices per equation (6). Over the course of time continual updating of the Euler transformation can be done by matrix multiplying by a succession of incremental transformation matrices in a chain-serial matrix multiplication procedure.
A conventional method used to stabilize detections is to mount the sensor on a stabilized platform. The platform is gimbaled to permit rotations around yaw, pitch and roll axes; and the position of the platform is maintained by three gyros. Electrical outputs of the gyros are proportional to the angles (or rates) through which the gimbals have moved. These gyro angles represent order-dependent rotations about each of the gimbal axes. In other words, these angles represent a rotation about the yaw axis, followed by a rotation about the pitch axis, followed by a rotation around the roll axis. These are the Euler angles of rotation, which can be used to form the Euler transformation matrices.
For practical reasons detection systems often need to be hard-mounted to aircraft hulls as opposed to being mounted on stabilized platforms. Sensor data therefore includes the effects of any aircraft motion. To reduce the system false alarm rate to an acceptable level, potential threats must be tracked in an inertial coordinate system. Effects of aircraft motion must therefore be eliminated from the raw sensor data before tracking can be done. Three rateintegrating gyros located at the sensor and strapped down to the aircraft hull will be able to measure motion caused by aircraft maneuvers as well as motion caused by aircraft vibrations, flexure and turbulence.
The angles measured by the strapped down gyros are not the Euler angles of rotation. In the prior art the Euler angles have been calculated proceeding with three simultaneous differential equations relating the gyro yaw, pitch and roll rates, and changing them to Euler rates. These are then integrated to form Euler angles. Then the Euler angles have been used to form a transformation matrix [T] per equation (4). This involves a large number of calculations.
The invention avoids making so many calculations by deriving an incremental transformation matrix more directly, relying on small angle approximations. A first-order differential equation as set forth in equation (7), following, can be solved to find the transformation matrix T(t). EQU T(t)=T(t)W(t) (7)
The transformation matrix T(t), is a function of time, and so is the coefficient matrix W(t). The coefficient matrix W(t) is defined as follows. ##EQU6##
The elements of the W matrix are the rates measured by the three gyros.
The solution to the differential equation (7) is EQU T(t)=T(t.sub.o).PHI.(t,t.sub.o) (9)
where (t,to) is the transition matrix associated with the equation, given by ##EQU7## (Refer to R.W. Brockett, Finite Dimensional Linear Systems, John Wiley & Sons, N.Y., 1970, pp.19-23).
If .tau.=t-t.sub.o is small, i.e., .vertline..tau..vertline.&gt;&gt;1, then W(.sigma.) is approximately constant for all t.sub.o .ltoreq..sigma..ltoreq.t. In this case .phi.(t,t.sub.o) can be approximated as follows: ##EQU8## Truncating second-order and higher terms, we obtain the following: ##EQU9## where I is the unit matrix and .alpha.,.beta.,.UPSILON. are the angles each gyro rotates through during the time .tau.. Therefore, to update a transformation T(t.sub.1) to T(t.sub.2) where t.sub.2 -t.sub.1 =.tau., the following matrix multiplication is performed EQU T(t.sub.2)=T(t.sub.1)T (13)
where the incremental transformation matrix T is given by ##EQU10##